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Exponential random graphs and dynamic graph algorithms

Exponential random graphs and dynamic graph algorithms. David Eppstein Comp. Sci. Dept., UC Irvine. What are we trying to do?. Probabilistic reasoning on social networks Appropriately model the different likelihoods of finding different types of network. Exponential Random Graphs.

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Exponential random graphs and dynamic graph algorithms

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  1. Exponential random graphsanddynamic graph algorithms David Eppstein Comp. Sci. Dept., UC Irvine

  2. What are we trying to do? • Probabilistic reasoning on social networks • Appropriately model the different likelihoods of finding different types of network

  3. Exponential Random Graphs • Family of graphs with fixed vertex set • Probability of a graph is proportional to exp(sum of weights of features) • Different choices of features give simpler or more powerful models

  4. ERG models can be simple… • Easily subsumes many standard random graph models • E.g. G(n,p): • Edges are independent w/probability p • Feature = edge • Weight = log(p) - log(1-p)

  5. …but ERG models can also be very powerful • Powerful enough to represent any distribution over n-vertex graphs • Feature = isomorphism with one graph • Weight = log(probability of that graph) • More power requires a more complex set of features

  6. Computational tasks for reasoning with ERGs • Compute normalizing factor (partition function) for graph probabilities • Generate random graphs from the model • Use the model as a prior for max-likelihood data fitting, or modify the feature weights to fit the data

  7. Monte Carlo methods for computing with ERGs • Start with an arbitrary graph • Repeatedly propose a small change (e.g., insert or delete a single edge) • Compute log-likelihood of the modified graph and use it to accept or reject the proposed change

  8. The Algorithmic Lens • Social scientists and statisticians determine the sorts of models that best describe their data • Algorithms researchers (e.g. me) figure out how to make the model run quickly • Faster algorithms lead to the ability to use more accurate models

  9. Algorithmic rephrasing of the computational task • Maintain a dynamic graph subject to edge insertions and deletions • As the graph changes, keep track of its computational properties efficiently (faster than recomputing them from scratch) • The properties we track should be the ones needed for ERG feature vectors

  10. A brief survey of dynamic graph algorithms • Sparsification (E., Galil, Italiano, Nissenzweig, JACM ‘92): • Replace dense graphs by tree of sparse subgraphs • Applies to many problems including maintaining connected components • Replaces #edges by #vertices in running times of update algorithms

  11. A brief survey of dynamic graph algorithms • Fast dynamic connectivity (Holm, de Lichtenberg, Thorup, JACM 2001): • Maintain connected components, number of connected components, or a spanning tree (so can use #components as ERG feature) • Update time O(log n log log n) • Complicated, of interest to search for more easily implemented variants

  12. A brief survey of dynamic graph algorithms • Distance and reachability in graphs • Of likely use in ERGs (e.g. to model small-world properties of these graphs) • Some dynamic graph algorithms are known but more theoretical than practical

  13. A brief survey of dynamic graph algorithms • Graphs in the plane and on surfaces • E. et al, J. Algorithms 1992 • E. et al, J. Comp. Sys. Sci. 1996 • E., SODA 2002 • Of possible interest for integrating social networks with geographic data

  14. Not-yet-dynamized graph algorithms useful for ERGs • Low-degree orientations of sparse graphs (Chrobak, E., Theor. Comp. Sci. 1991) • Assign directions to the edges of the graph so that each vertex has O(1) outgoing edges • Enable fast search for small subgraphs (e.g. list all cliques in linear time) • May be found in linear time

  15. Not-yet-dynamized graph algorithms useful for ERGs • Finding all maximal complete bipartite subgraphs in a sparse graph (E., IPL 1994) • Allows concise representation of all four-vertex cycles (quadratically many cycles may be represented in linear space and time) • Based on low-degree orientation

  16. Not-yet-dynamized graph algorithms useful for ERGs • Subgraph isomorphism: finding all copies of some small pattern graph in a larger graph (E., J. Graph Th. 1993 and J. Graph Algorithms 1999) • Commonly used as ERG features • Known fast algorithms rely on special graph properties e.g. planarity

  17. Not-yet-dynamized graph algorithms useful for ERGs • Centrality: measuring the importance of a node in a social network • Fast approximation of closeness centrality based on small world hypothesis (E., Wang, J. Graph Algorithms 2004) • Other centralities e.g. betweennness

  18. Conclusions • ERG are important model for social nets • ERG computation naturally involves dynamic graphs • Many existing dynamic graph algorithms known, not fully adapted to ERG problems • Much opportunity for further study of dynamic graph algorithms in ERG setting

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